In a noise driving by a multivariate point process $\mu$ with predictable compensator $\nu$, we prove existence and uniqueness of the reflected backward stochastic differential equation's solution with a lower obstacle $(\xi_{ t})_{ t\in[0,T]}$ which is assumed to be right upper-semicontinuous but not necessarily right-continuous process and a Lipschitz driver $f$. The result is established by using Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of It\^{ o}'s formula due to Gal'chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.
Categories: math.PR